Can one get infected by a sexually transmitted disease if one ingests bodily fluids from an infected person?

Yes.... sort of: Oral sex is one way to pass std. Herpes, hpv and syphilis are passed by mouth to skin contact. Gonorrhea, chlamydia and HIV are examples of disease passed by fluids. Ingesting the fluid though is not the issue. The fluid coming into contact with the tissue of the mouth and throat are how infections occur.

Very unlikely: If the "bodily fluid" in question is semen or other sexual secretions (e.g. cervical-vaginal fluid), there could be risk of gonorrhea or other STDs. No other "bodily fluid" is risky for STDs, and that includes blood. For example, oral exposure to HIV -- even infected blood -- rarely result in infection.